Subgroups — #35295699

Order 21248.

374 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14581728	C₂	1	1	yes	maximal, characteristic
2	#24905744	C₂	1	2	yes	maximal
3	C₂×C₈×C₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#24905299	C₂	1	2	yes	maximal
5	C₈₃×#1564	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581728	C₄	1	2	yes	—
7	C₄×#13050574	C₂²	1	2	yes	—
8	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
10	#19127466	C₄	1	2	yes	—
11	#19127423	—	2	2	no	—
12	C₈×C₈₃⋊C₈	—	2	2	no	—
13	C₈×C₈₃⋊C₈	C₄	1	2	yes	—
14	C₈×C₈₃⋊C₈	C₄	1	2	yes	—
15	#431×C₈₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	#335×C₈₃	C₄	1	2	yes	—
18	#334×C₈₃	C₄	1	2	yes	—
19	C₂×#13050574	C₂×C₄	1	4	yes	—
20	C₂²×C₈₃⋊C₈	—	2	1	no	—
21	C₂²×C₈₃⋊C₈	Q₈	1	2	yes	—
22	#14581728	C₈	1	8	yes	complemented
23	#14581703	—	2	2	no	—
24	C₄×C₈₃⋊C₈	—	2	2	no	—
25	C₄×C₈₃⋊C₈	—	2	1	no	—
26	C₄×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
27	C₄×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
28	C₄×C₈₃⋊C₈	D₄	1	1	yes	characteristic
29	C₄×C₈₃⋊C₈	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₈₃	C₂×C₄	1	4	yes	—
34	C₂²⋊C₈×C₈₃	—	2	2	no	—
35	C₄×C₈×C₈₃	—	2	2	no	—
36	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
40	#13050574	—	2	4	no	—
41	#13050574	C₂×C₈	1	8	yes	—
42	C₂×C₈₃⋊C₈	—	2	4	no	—
43	C₂×C₈₃⋊C₈	—	2	2	no	—
44	C₂×C₈₃⋊C₈	—	2	4	no	—
45	C₂×C₈₃⋊C₈	—	2	4	no	—
46	C₂×C₈₃⋊C₈	C₄⋊C₄	1	2	yes	—
47	C₂×C₈₃⋊C₈	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
51	C₂×C₈×C₈₃	—	2	2	no	—
52	C₂×C₈×C₈₃	—	2	4	no	—
53	C₂×C₈×C₈₃	C₄²	1	1	yes	characteristic
54	C₂×C₈×C₈₃	C₄²	1	1	yes	characteristic
55	C₂×C₈×C₈₃	C₄²	1	1	yes	characteristic
56	C₂×C₈×C₈₃	C₄²	1	1	yes	characteristic
57	C₂×C₈×C₈₃	C₂×C₈	1	4	yes	—
58	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₈₃	#41	1	1	yes	characteristic
62	C₂³×C₈₃	C₄×Q₈	1	1	yes	characteristic
63	C₂×C₄×C₈₃	—	2	2	no	—
64	C₂×C₄×C₈₃	—	2	2	no	—
65	C₂×C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
68	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₈₃	C₂²×C₈	1	2	yes	—
72	C₂×C₄×C₈₃	#131	1	2	yes	—
73	C₈×C₈₃	—	2	1	no	—
74	C₈×C₈₃	—	2	1	no	—
75	C₈×C₈₃	—	4	4	no	—
76	C₈×C₈₃	C₄×C₈	1	2	yes	—
77	C₈×C₈₃	C₄×C₈	1	2	yes	—
78	C₈₃⋊C₈	—	2	4	no	—
79	C₈₃⋊C₈	—	2	2	no	—
80	C₈₃⋊C₈	—	2	8	no	—
81	C₈₃⋊C₈	#108	1	4	yes	complemented
82	C₂²×C₈₃	—	2	1	no	—
83	C₂²×C₈₃	—	2	1	no	—
84	C₂²×C₈₃	C₄×C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₈₃	C₈×Q₈	1	2	yes	—
86	C₄×C₈₃	—	2	1	no	—
87	C₄×C₈₃	—	2	1	no	—
88	C₄×C₈₃	C₂×C₄×C₈	1	1	yes	characteristic
89	C₄×C₈₃	#405	1	1	yes	characteristic
90	C₄×C₈₃	C₂×#108	1	2	yes	—
91	C₄×C₈₃	C₈×D₄	1	2	yes	—
92	C₄×C₈₃	#443	1	2	yes	—
93	C₈×#108	—	83	1	no	maximal, Sylow
94	C₂×C₈₃	—	2	2	no	—
95	C₂×C₈₃	C₈×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₈₃	#1772	1	1	yes	characteristic
97	C₂×C₈₃	C₄×#108	1	1	yes	characteristic
98	C₄×#108	—	83	1	no	—
99	#1564	D₈₃	1	1	yes	characteristic
100	C₈×M₁₆	—	83	2	no	—
101	C₂×C₈²	—	83	1	no	—
102	#1278	—	83	2	no	—
103	C₈₃	C₈×#108	1	1	yes	characteristic, complemented, Sylow
104	#431	C₂×D₈₃	1	2	yes	—
105	C₂×#108	—	83	2	no	—
106	C₄×M₁₆	—	83	2	no	—
107	C₂×C₄×C₈	—	83	1	no	—
108	C₂×C₄×C₈	—	83	1	no	—
109	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
110	#341	—	83	2	no	—
111	#335	Dic₈₃	1	2	yes	—
112	#334	Dic₈₃	1	2	yes	—
113	#322	—	166	2	no	—
114	C₈²	—	83	2	no	—
115	C₈²	—	83	2	no	—
116	C₈²	—	166	2	no	—
117	C₂×M₁₆	—	83	4	no	—
118	C₂²×C₈	—	83	2	no	—
119	C₂²×C₈	—	166	1	no	—
120	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
121	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
122	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
123	#108	—	83	8	no	—
124	C₄⋊C₈	C₂×Dic₈₃	1	4	yes	—
125	C₂²⋊C₈	—	2	2	no	—
126	#97	—	166	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	83	2	no	—
129	C₄×C₈	—	83	2	no	—
130	C₄×C₈	—	83	1	no	—
131	C₄×C₈	—	83	1	no	—
132	C₄×C₈	—	166	2	no	—
133	C₄×C₈	—	166	1	no	—
134	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
135	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
136	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
137	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
139	C₂²×C₄	Q₈×D₈₃	1	1	yes	characteristic
140	C₂²×C₄	#13050589	1	1	yes	characteristic
141	M₁₆	—	83	8	no	—
142	M₁₆	—	166	4	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	—	83	2	no	—
146	C₂×C₈	—	83	2	no	—
147	C₂×C₈	—	166	4	no	—
148	C₂×C₈	—	166	4	no	—
149	C₂×C₈	—	166	2	no	—
150	C₂×C₈	—	166	4	no	—
151	C₂×C₈	C₄×Dic₈₃	1	1	yes	characteristic
152	C₂×C₈	C₄×Dic₈₃	1	1	yes	characteristic
153	C₂×C₈	C₄×Dic₈₃	1	1	yes	characteristic
154	C₂×C₈	C₄×Dic₈₃	1	1	yes	characteristic
155	C₂×C₈	C₈×D₈₃	1	4	yes	—
156	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
157	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
158	C₄²	D₄×D₈₃	1	1	yes	characteristic
159	C₄²	#13050591	1	1	yes	characteristic
160	C₂³	#14581757	1	1	yes	characteristic
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	C₂×C₄×Dic₈₃	1	1	yes	characteristic
164	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
165	C₂×C₄	#14581750	1	1	yes	characteristic
166	C₂×C₄	#14581754	1	1	yes	characteristic
167	C₂×C₄	D₄×Dic₈₃	1	1	yes	characteristic
168	C₂×C₄	Q₈×Dic₈₃	1	1	yes	characteristic
169	C₂×C₄	C₂×C₈×D₈₃	1	2	yes	—
170	C₂×C₄	#14581800	1	2	yes	—
171	C₈	—	2	1	no	—
172	C₈	—	2	1	no	—
173	C₈	—	4	4	no	—
174	C₈	—	83	4	no	—
175	C₈	—	166	4	no	—
176	C₈	—	166	2	no	—
177	C₈	—	166	8	no	—
178	C₈	C₈×Dic₈₃	1	2	yes	—
179	C₈	C₈×Dic₈₃	1	2	yes	—
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	C₄⋊C₄×Dic₈₃	1	1	yes	characteristic
183	C₂²	#19127825	1	2	yes	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	C₂×C₈×Dic₈₃	1	1	yes	characteristic
187	C₄	#19127659	1	1	yes	characteristic
188	C₄	#108×D₈₃	1	2	yes	—
189	C₄	#19127766	1	2	yes	—
190	C₄	#19127818	1	2	yes	—
191	C₂	—	2	2	no	—
192	C₂	#24907328	1	1	yes	characteristic
193	C₂	#24907344	1	1	yes	characteristic
194	C₂	#108×Dic₈₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₈₃
commutator	C₄×C₈₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1564	C₂
socle	C₂²×C₈₃	C₄×C₄⋊C₄
prime radical	C₂³×C₈₃	C₄×Q₈
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1564	D₈₃
hypocenter	C₈₃	C₈×#108
LCS	G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1564

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
22. #14581728	C₈	semidirect
81. C₈₃⋊C₈	#108	semidirect
103. C₈₃	C₈×#108	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×#14581728	2
#24905744	2
C₂×C₈×C₈₃⋊C₈	2
#24905299	2
C₈₃×#1564	2
C₈×#108	83